AN ALMOST SURE KPZ RELATION FOR SLE AND BROWNIAN MOTION

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Holden, Nina; Miller, Jason
• 署名单位: University of Cambridge; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1385
• 发表日期: 2020
• 页码: 527-573
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS; QUANTUM-GRAVITY; intersection exponents; Hausdorff Dimension; SCALING LIMITS; arm exponents; thick points; random-walks; planar maps; Duality

摘要

The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a gamma-Liouville quantum gravity (LQG) surface, gamma is an element of(0, 2), decorated with a space-filling form of Schramm's SLE kappa(,) kappa = 16/gamma(2) is an element of (4, infinity), eta as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion Z. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset A of the range of eta, which can be defined as a function of eta (modulo time parameterization) to the Hausdorff dimension of the corresponding time set eta(-1) (A). This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an SLE, CLE or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLE kappa curve for kappa not equal 4; the double points and cut points of SLE kappa for kappa > 4; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of m-tuple points of space-filling SLE kappa for kappa > 4 and m >= 3 by computing the Hausdorff dimension of the so-called (m - 2)-tuple pi/2-cone times of a correlated planar Brownian motion.